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Photonic Crystals: Periodic Surprises in Electromagnetism

Photonic Crystals: Periodic Surprises in Electromagnetism. Steven G. Johnson MIT. scattering. planewave. To Begin: A Cartoon in 2d. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •.

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Photonic Crystals: Periodic Surprises in Electromagnetism

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  1. Photonic Crystals:Periodic Surprises in Electromagnetism Steven G. Johnson MIT

  2. scattering planewave To Begin: A Cartoon in 2d

  3. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • planewave ...but for some l (~ 2a), no light can propagate: a photonic band gap To Begin: A Cartoon in 2d a for most l, beam(s) propagate through crystal without scattering (scattering cancels coherently)

  4. 1887 1987 (need a more complex topology) Photonic Crystals periodic electromagnetic media with photonic band gaps: “optical insulators”

  5. can trap light in cavities and waveguides (“wires”) magical oven mitts for holding and controlling light Photonic Crystals periodic electromagnetic media with photonic band gaps: “optical insulators”

  6. But how can we understand such complex systems? Add up the infinite sum of scattering? Ugh! Photonic Crystals periodic electromagnetic media

  7. + + + + + current: conductivity (measured) mean free path (distance) of electrons A mystery from the 19th century conductive material e– e–

  8. 10’s of periods! + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + current: conductivity (measured) mean free path (distance) of electrons A mystery from the 19th century crystalline conductor (e.g. copper) e– e–

  9. electrons are waves (quantum mechanics) 1 waves in a periodic medium can propagate without scattering: Bloch’s Theorem (1d: Floquet’s) 2 The foundations do not depend on the specificwave equation. A mystery solved…

  10. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • planewave Time to Analyze the Cartoon a for most l, beam(s) propagate through crystal without scattering (scattering cancels coherently) ...but for some l (~ 2a), no light can propagate: a photonic band gap

  11. + constraint eigen-state eigen-operator eigen-value Fun with Math First task: get rid of this mess 0 dielectric function e(x) = n2(x)

  12. + constraint eigen-state eigen-operator eigen-value Hermitian Eigenproblems Hermitian for real (lossless) e well-known properties from linear algebra: w are real (lossless) eigen-states are orthogonal eigen-states are complete (give all solutions)

  13. Periodic Hermitian Eigenproblems [ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ] [ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ] if eigen-operator is periodic, then Bloch-Floquet theorem applies: can choose: planewave periodic “envelope” Corollary 1: k is conserved, i.e.no scattering of Bloch wave Corollary 2: given by finite unit cell, so w are discrete wn(k)

  14. ? range of k? Periodic Hermitian Eigenproblems Corollary 2: given by finite unit cell, so w are discrete wn(k) band diagram (dispersion relation) w3 map of what states exist & can interact w2 w w1 k

  15. Consider k+2π/a: periodic! satisfies same equation as Hk = Hk k is periodic: k + 2π/a equivalent to k “quasi-phase-matching” Periodic Hermitian Eigenproblems in1d e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e(x) = e(x+a) a

  16. band gap Periodic Hermitian Eigenproblems in1d k is periodic: k + 2π/a equivalent to k “quasi-phase-matching” e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e(x) = e(x+a) a w k 0 –π/a π/a irreducible Brillouin zone

  17. Any1d Periodic System has a Gap [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Start with a uniform (1d) medium: e1 w k 0

  18. bands are “folded” by 2π/a equivalence Any1d Periodic System has a Gap [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Treat it as “artificially” periodic e1 e(x) = e(x+a) a w k 0 –π/a π/a

  19. Any1d Periodic System has a Gap [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Treat it as “artificially” periodic a e(x) = e(x+a) e1 w 0 π/a x = 0

  20. a e(x) = e(x+a) e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 Any1d Periodic System has a Gap [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Add a small “real” periodicity e2 = e1 + De w 0 π/a x = 0

  21. a e(x) = e(x+a) e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 Any1d Periodic System has a Gap [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Splitting of degeneracy: state concentrated in higher index (e2) has lower frequency Add a small “real” periodicity e2 = e1 + De w band gap 0 π/a x = 0

  22. Some 2d and 3d systems have gaps • In general, eigen-frequencies satisfy Variational Theorem: “kinetic” inverse “potential” bands “want” to be in high-e …but are forced out by orthogonality –> band gap (maybe)

  23. algebraic interlude algebraic interlude completed… … I hope you were taking notes* [ *if not, see e.g.: Joannopoulos, Meade, and Winn, Photonic Crystals: Molding the Flow of Light ]

  24. a 2d periodicity, e=12:1 frequencyw(2πc/a) = a / l irreducible Brillouin zone G G X M M E gap for n > ~1.75:1 TM X G H

  25. Ez – + 2d periodicity, e=12:1 Ez (+ 90° rotated version) G G X M E gap for n > ~1.75:1 TM H

  26. a 2d periodicity, e=12:1 frequencyw(2πc/a) = a / l irreducible Brillouin zone G G X M M E E TM TE X G H H

  27. 2d photonic crystal: TE gap, e=12:1 TE bands TM bands E TE H gap for n > ~1.4:1

  28. 3d photonic crystal: complete gap , e=12:1 I. II. gap for n > ~4:1 [ S. G. Johnson et al., Appl. Phys. Lett.77, 3490 (2000) ]

  29. You, too, can computephotonic eigenmodes! MIT Photonic-Bands (MPB) package: http://ab-initio.mit.edu/mpb on Athena: add mpb

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